Pythagorean Triples

I got interested in this problem last summer at the Math Circle Institute, held at Notre Dame. I have a bad memory and am having fun reconstructing what we figured out together there.


Introduction to Pythagorean Triples
The Pythagorean Theorem tells us that a right triangle with legs a and b and hypotenuse c will always have the relationship a²+b² = c². (Do you know how to prove that?) When all three sides are whole numbers, we have a Pythagorean triple. The most famous of these is 3²+4² = 5², often referred to in this context as (3,4,5). The 3-4-5 triangle was used in Egypt to help make perpendicular sides for their magnificent buildings. A loop of rope with 12 equally spaced knots (3+4+5 = 12) was pulled taut at knots 0, 3 and 7 to make a precise right angle.

If the three sides don't all have a factor in common, then they make a primitive Pythagorean triple (PPT). 6²+8² = 10² is not a PPT because all three sides have a factor of 2.


Starting to Explore
When I'm making up problems for students, I often want another Pythagorean triple, so I have one more sitting in my brain, 5²+12² = 13². Are there others? Are there infinitely many PPT's? How would we find more?

One approach to exploring these involves thinking about parity. (Parity refers to whether a number is odd or even.) I see that both of the PPT's above, (3,4,5) and (5,12,13), are odd+even = odd. Can we have odd+odd = even? What about even+even = even? If we have odd+even = odd, does the odd leg have to be the shorter one?

Do other questions occur to you?

Recommendation: stop reading and start playing as soon as you have a thought about how you might proceed.

I knew there were more PPT's, but couldn't remember any others. I wanted to find a few more, so I could see any obvious patterns. So I made a list of the first 25 perfect squares and looked for pairs that would add to equal another perfect square, or subtract to equal another one. I found (8,15,17) and (7,24,25). Well, that's one question answered: the odd leg does not have to be the shortest side. I see that all 4 hypotenuses are odd. So I want to address the question of whether odd+odd = even is possible.


Question1: Is odd+odd = even possible?
Here's how I start: Suppose we have two odd legs. We can let a = 2n+1 and b=2m+1. Then a²+b² = ... Can we come to a contradiction? [See the hint at the end for a bit more direction.]


Question2: If a is an odd number, can I find a PPT for it?
I notice that all 3 triples in which the odd leg is the short one include consecutive numbers for the other leg and the hypotenuse. Hmm. If a is the short leg, then b+c = a² in all 3 of those cases. Is that important? What if I write a²+b² = c² as c²-b² = a²? Oh! A square minus a square can factor. So I'd have (c-b)(c+b) = a². What does that get me? [I found a way to get a PPT for every odd number. Can you?]


Question3: Must the even side be a multiple of 4?
I wanted more triples, in case it would help me see more patterns. So I set up a spreadsheet with column a holding 1 through 100, row 1 holding 1 through 254 (first time I've ever used all available rows!). Column b and row 2 had the squares of these numbers. The rest of the spreadsheet showed a 0 if the square root of the sum of these squares was not a whole number, and otherwise showed the number. [Here's the formula in cell c3: =IF(INT(SQRT($B3+C$2))=SQRT($B3+C$2),SQRT($B3+C$2),0)]

For most multiples of 4, I found a PPT. And I didn't find any for the other even numbers. So I wanted to know whether the even side had to be a multiple of 4. If the even side is a, then b and c are odd, and c²-b², with b=2m+1 and c=2n+1, can be explored.


Questions 4 and 5: Multiples of 3, 4 and 5
This reminded me that I had read (whose blog was that on?) that in every PPT, 3 will be a factor of one side, 4 will be a factor of one side, and 5 will be a factor of one side. (As in (5,12,13), one side may contain more than one of these factors.) I realized I'd already proved it for 4. I started trying to prove it for 3.

I mentioned parity earlier. Even numbers can be expressed as a=2m, and odd numbers can be expressed as b=2n+1. Similarly, if we want to think about whether side c will be a multiple of 3, we can look at three cases: c=3m, c=3m+1, or c=3m+2. Using this, I started with the question of whether c would be a multiple of 3. (It wasn't in any of the triples I'd found.) If it is, then neither a nor b can be. (Why?) Once I solved that problem, I wanted to prove that one of the legs would be a multiple of 3. Suppose b is not a multiple of 3 and consider c²-b². There will be 4 cases, each of b and c can either be 3x+1 or 3x+2. What does this make a?

I tried to think about 5 in this way but got nowhere. I'm writing this blog post in hopes that explaining my thinking will help me get further on some of my dead ends.

I have a few other questions I haven't answered:

• Given a multiple of 4, how can I come up with a PPT?
• Can the same number show up in 3 different PPT's?
  

Let me know if you have fun playing with this. Maybe the directions your thinking takes will be different than mine...

Hint: c² is a multiple of 4. (Why?) What about a²+b²?

Math Teachers at Play #21

Welcome once again ...
to the Math Teachers At Play blog carnival. Puzzlers, riddlers, thinkers, doers, novices, experts, come one , come all!

[photo by santarosa]

First off, in honor of the number 21, is a puzzle, fresh from the oven.

The Numberland News runs personal ads. 21 was looking for a new friend and put an ad in.

Two-digit, semi-prime, triangular, Fibonacci number seeks same. I'm a binary palindrome, what about you?

Will 21 find a friend?


Elementary
Kendra at Aussie Pumpkin Patch has written about the estimation lesson she did with her sons, which they started by reading Counting on Frank. It sounds like fun!

What is your child's favorite small toy? Will it help them learn division? Ashley at HyperHomeschool headed for the Legos, and here's what happened.

A really good way to understand place value is to work with other number bases. A book I recently discovered, How to Count Like a Martian, by Glory St. John, tells a detective story in which the history of other number systems plays a starring role. The last few chapters discuss place-value-based systems. Want a more hands-on approach? Sol, at Wild About Math!, offers us some math magic with index cards, based on binary numbers.

Megan Wong has written some books she'd like to share with us: Math Power is Fun and Brain Power is Fun are part of her Mind Power series.


Algebra and Geometry
At Let's Play Math, Denise gives another algebra lesson in Pre-Algebra Problem Solving: 4th Grade. I'm thinking this would be a good first step even for adults just learning algebra. I've seen mention of bar diagrams plenty, but this is the best illustration of their use I've seen so far.

Maria Miller offers us 3 videos of her proofs of some basic geometric relationships in Angles in a parallelogram and a triangle.

John Golden at Math Hombre shares his geogebra sketches (available as webpages and geogebra files) at Net Results. Students can use these to create their own prisms and pyramids. Print the nets, fold them up and see the funky solids.

One of my favorite things about MTAP is discovering new blogs. Here's one: Guillermo presents a Tutorial on Geometer's Sketchpad.


Calculus and ...
Pat's Blog has Fun with Parabolas.

Dan at Mathrecreation offers us a curious population model, with directions for exploring the logistics functions in Fathom.


Our Favorite Proofs
Brent, at The Math Less Traveled, presents a proof that pi is irrational. He thinks calculus students should be able to follow it. I've treid to follow this proof in other places and not had the patience for it. So far, his explanation is right up my alley.

The Count, at Discrete Ideas, gives us Discretely Simple, on his two favorite proofs.


On Teaching
Simple things can make such a difference. “What’s a question that someone else might get wrong?” So simple, and such a good way to get students thinking. Here's JD's post on it.

Riley Lark, at Point of Inflection, offers us his index cards. Well, the students get the cards, and some quickie interaction.

Then they can review with Trashketball. Post by Dan Greene at The Exponential Curve.


News

• TI and Numb3rs
• DARPA's red balloon project
• TIMSS Advanced (from Math Education Research Blog)
• I wrote to Ivar Ekeland, author of The Cat in Numberland, and he wrote back! He says the publisher is "preparing a re-issue in 2010". Yeay! I think I'll be getting lots of copies of it next year around this time to give as presents.
  

The Holiday Connection
When mathematicians hear about the gifts "my true love gave to me" on the 12 days of Christmas, they start counting. How many gifts would that be altogether? Sol at Wild About Math! wrote this post a few years back. And John at The Endeavor wrote this post more recently. One of the commenters on John's post wondered: "Funny that the 12 days of Christmas turn out to have just short of one gift per day for a full year. A coincidence?"

Remember the Soma Cube? Rachel, at Minds in Bloom, gives directions for making it here, and thinks a home-made puzzle would make a great gift. If you'd like to do that for this holiday season you may not have time to wait for the cubes to be shipped. But if you can wait, the cubes are pretty inexpensive online: 100 1" plastic cubes on Amazon for about $15, 100 1" wooden cubes for about $10 here, or 1000 centimeter cubes for $25.

Between Maria and JD and a few others around here, I've started to think that creating puzzles (or authoring math, as Maria would say) is something I too can do. So last week I made a logic puzzle, Holiday Logic. I hope you'll enjoy it.




This edition of MTAP was composed in Richmond, California and Chicago, Illinois. It's coming out late in the day because leaving home and flying here yesterday, even though uneventful, did take up my whole day. May your holidays be peaceful. May peace spread exponentially from our hearts through our actions to the world around us.

Holiday Logic Puzzles

I'd like to share two puzzles here. One is by Mike Shenk, who has a site called puzzability, and is interviewed here. It's called Oh Deer!, and it's a killer. The other I created last week. I wanted a logic puzzle with a holiday theme for my math salon, and I wanted it to be much easier than Oh Deer! When I searched online, I got tired of seeing so much about buying presents. I also wanted to include other holidays besides xmas.

I've always loved doing these puzzles, but this is my first attempt at creating one. I believe it has a unique answer. Please let me know if I goofed.

I wish you all peace and joy during these winter holidays.

Holiday Logic
by Sue VanHattum

1. The Green girl’s favorite Christmas tradition is singing carols.
2. The Brown boy celebrates Kwanzaa indoors.
3. DJ and Jordon joined their friend in her candle lighting ceremony.
4. Layla and Amani joined their friend for his annual walk through the woods.
5. The Gold girl came to Jordon’s house to join his family in their feast.
6. The Fox family celebrate the Yuletide, and Amani comes to their party.
7. Amani couldn’t make it to the Gold family’s Hanukkah celebration.

[Edited for clarity.] Each child celebrates just one holiday with one special activity as a tradition in their family, though they do join in the fun with their friends this year. Your mission: Decide who celebrates each holiday, and what they do to celebrate.

Oh Deer! A logic problem by Mike Shenk
(first published in Games Magazine, December 1992)

Twas the night before Christmas, and at the North Pole
The last-minute planning was taking its toll.
As Santa was hastily making a scheme
For the placement of deer in his sleigh-pulling team,
The good Mrs. Claus was crocheting bright bows
To be worn by these reindeer (four bucks and four does).

The ribbons were colored in eight festive hues:
One ocher, one rose, one cerise, one chartreuse,
One maroon, one magenta, one white, and one blue.
(These ribbons helped Santa keep track of who's who.)
The deer pulled the toy-laden sleigh in four rows,
Arranged so no row held two bucks or does.

The order of pullers was changed year by year,
For Santa was thoroughly fair with his deer.
He summoned the elves and instructed them thus:
"Let's hitch up the reindeer with minimum fuss.
The bow on the buck behind Dasher is white,
While Blitzen, a doe, sees cerise to her right.

The blue bow is nearer my sleigh than is Dancer,
But nearer the front of my team than is Prancer.
The doe in chartreuse gets a front-of-team honor,
But not on the same side as Cupid or Donner.
Now Comet stands two spots ahead of the rose.
And three deer of four on the right side are does.

The cerise bow is worn two in back of maroon,
One of which is beside the bright ocher festoon.
Oh-Cupid's in front of a buck, by the way.
Well, that's how they line up for pulling my sleigh.
I trust that you elves, being clever, now know
Each reindeer's position and color of bow."

In no time each colorful ribbon was tied
And the team was hitched up for the transglobal ride.
Can you ascertain where each member fits in?
Who's Comet? Who's Cupid" Where's Donner? And Blitzen?
Who's Dasher? Who's Dancer" Where's Vixen? And Prancer?
With logical thought, you'll determine the answer
And write down the color and place for each deer.
Happy Christmas to all, and to all much good cheer!

Hannah, Divided, by Adele Griffin

Over the past year I've learned a new term: 2e, or twice exceptional, is a term used by advocates of kids who are exceptionally smart, along with having exceptional learning differences. (I'm paraphrasing Tiffani, who blogs beautifully at Child's Play about 2e issues. Another blog I've enjoyed on the subject is Life Among the Gifted.)

Hannah, Divided is the sweet story of a girl who would be designated 2e nowadays. Growing up during the depression on a farm, she's not too worried about her struggles with reading, and the comfort she takes in numbers is very personal. She doesn't much expect either her learning trouble or her gift to take her away from milking cows and sharing the chores with her family. But they do.

Her teacher, Miss Cascade, has prepared the one-room schoolhouse and all its students for a visit from a possible benefactor. On the day Mrs. Sweet arrives, she takes an interest in Hannah's math abilities and quizzes her after school. On the way home, Hannah is so fired up, she just has to run, and count.

Finally, she took this year, 1934, and divided it by two. Over and over, skipping the decimal point like a checkers piece until it stood at the front of the line.

Granddad McNaughton encouraged her mathematics. Sunday afternoons, they passed gleeful hours inventing games with figures and sums, making up riddles and puzzles to solve.

She's given a chance to go to Philadelphia to study math, and takes it. Leaving home is difficult for her, and she leans on her need to pace her room 32 times, get each item in exactly the right place, and tap her paper 32 times. As hard as it is, the math she's able to learn from her tutor at Ottley Friends' School makes all the hardship worthwhile.

As I read this simple story, I kept thinking of how moving it might be for my young friend Artemis, and other kids like Hannah. I hope some of you have a chance to enjoy it soon.

Math Teachers at Play coming up...

Submissions are due in a week, on Wednesday, December 16. Please send your favorite posts that haven't been in MTAP yet, new or old. Send your own; send links to other people's posts that you like; send math questions that have you puzzled.

On Theorems and Proofs

There's a good discussion, over at f(t), on "What's your favorite theorem?" (It started on Twitter, but I like blogs better.)

And Brent (of The Math Less Traveled) has begun a lesson I've been longing for. π (that's pi) is irrational. I knew that. But I knew it in a way that doesn't count in math. I took it on authority. I've tried to look up the proof, and didn't have the patience for following what I saw. I'm confident Brent will walk us through it gently. I'm looking forward to this. Maybe this will be my favorite theorem, once I learn it. ;^)

What's my favorite theorem? Hmm, I like:

• Why the square root of 2 is irrational,*
• Rationals are countable and reals aren't (that's the one Kate explained so well at f(t)),
  
• Pythagorean Theorem,
  
• Fundamental Theorem of calculus,
• Infinity of primes,
• Angles in a triangle add to 180 degrees,
• The ones in linear algebra that all go together.

It's hard to pick just one. The angles one is nice, because you can show it by ripping off the corners of a triangle. I know, that's not a proof. But the proof parallels the torn paper demonstration nicely.



* I'm having a bad internet day. I couldn't find the site that makes pretty equations.

Liars, Truthtellers, and Octopuses*

Tanya Khovanova has translated some puzzles from Russian and created some herself. Now I want to try it.

Here's the intro and the easiest of the problems:

... our characters are genetically engineered octopuses. The ones with an even number of arms always tell the truth; the ones with an odd number of arms always lie. ... Not only do octopuses lie or tell the truth according to the parity of the number of their arms, it turns out that the underwater world is so discriminatory that only octopuses with six, seven or eight arms are allowed to serve Neptune. ... our octopuses who worked as guards at Neptune’s palace were conversing:

• The blue one said, “All together we have 28 arms.”
• The green one said, “All together we have 27 arms.”
• The yellow one said, “All together we have 26 arms.”
• The red one said, “All together we have 25 arms.”

How many arms does each of them have?

If you enjoyed that, go visit Tanya's blog and try the others.

Now it's my turn. Can I do it? My 4 all work at Neptune's also.

• Aqua says, "Turquoise has 6 arms."
• Turquoise says, "Blue has 7 arms."
• Blue says, "Green has 7 arms."
• Green says, "Aqua has 7 arms."
• Aqua says, "The truthtellers have the same number of arms as each other."
  

Not as elegant, but I think it works. (I'm up in the middle of the night, so my brain may not be functioning well enough to do this.) Your turn...



* Wikipedia explains why it's not 'octopi'.

Gifts for math lovers

I posted back in June about my favorite math books. Any of those would make a great gift. But I'm excited about a few books I've read recently, and wanted to share them here for those of you who like to give books as gifts. Both are biographies of mathematicians.

Carry On, Mr. Bowditch, by Jean Lee Latham (1955) is written for younger readers, but will charm many adults too. It's a fictionalized account of the life of Nathaniel Bowditch, who loved math, but had to leave school when his family needed his help. He was indentured to a ship chandlery for 9 years, which dashed his hopes of someday going to Harvard to study math. But he spent his spare time learning everything he could on his own - he learned Latin so he could read Newton's Principia Mathematica, and then learned French so he could read another book recommended to him.

After his indenture ended, he sailed with a merchant ship, and became interested in the mathematics of navigation. He was incensed at the errors in the book of tables used for navigation, and began the laborious work of correcting them.

"You don't 'cast your eye' over navigation tables!" Nat barked. "When I checked that one table of Maskelyne's, I worked every figure three times, just to be sure I was right!"
"Three times? Every figure? But why in ..."
"Why not? Nat roared. "Mathematics is nothing if it isn't correct! Men's lives depend on those figures!" (page 161)

He also taught the crews how to "do a lunar", a startlingly egalitarian action back then. In this passage he's talking to a young woman who later becomes his wife, but this is much like the lessons he gave the crews he sailed with:

Nat said, "That's the North Star. If you think of the North Star as the middle of your clock face, and the line from it through those other stars as the hour hand, you can tell time."
"It says about one o'clock. Is that right?"
"No, this clock runs backwards."
"Is it eleven o'clock?"
"No, there's one other difference. It takes twenty-four hours for the Big Dipper to swing around the North Star. So every hour space on the clock face stands for two hours." (page 85)

Bowditch eventually decided to write his own book, which he hoped would be error-free, and would also include navigation lessons and general information needed by sailors. His book, the American Practical Navigator, published in 1902, is still carried on every U.S. naval vessel (according to Wikipedia).

Bowditch was born in 1773 in Salem, Massachusetts. I enjoyed reading about the early days of the U.S. as an independent nation from his perspective. My only concern with a book like this is that I'll mix up what's fiction and what's true. The astronomy lesson I've quoted is a bit oversimplified, but apparently close.


The Man Who Knew Infinity: A Life of the Genius Ramanujan, by Robert Kanigel, would be unbelievable if it were fiction or even slightly fictionalized. A friend of mine, who works with gifted kids, describes their learning style as sitting in the eye of a hurricane and grabbing at the ideas whirling by. As I read about Ramanujan's mathematical discoveries, I keep coming back to that image. Other mathematicians who worked with him were astounded by his process. Bruce Berndt noted that, although Ramanujan's proofs were often full of holes, his results were almost always correct, and suggested, "We might allow our thoughts to occasionally escape from the chains of rigor, and, in their freedom, to discover new pathways through the forest." (page 183)

Ramanujan grew up in South India and attended school sporadically. (In his younger years, he preferred learning on his own. Later, he couldn't deal with exams in subjects outside mathematics, and was kicked out of university.) It took him years of working as a clerk to support himself before he managed to catch the attention of a famous mathematician in England, G.H. Hardy, whose interest in him suddenly changed his life. He went from just scraping by with a job he had little interest in, to a paid position as a research student in mathematics at Presidency College in Madras, India. A year later he would sail to England to begin with Hardy the work of making his mathematical results comprehensible to others.

I'm only halfway through, but have learned much already about India, mathematics at Cambridge in the late 1800's, and the history of mathematics. Kanigel does a good job of giving us enough background so that we have some chance of understanding different cultures and different times. (At times, his own bias shows, but subtly.) His description of Hardy's writing (readable, clear, cogent, almost suspenseful) makes me want to go to my local math library and borrow his 1908 Course in Pure Mathematics.

One of the delights for me is the sweet trivia I'm picking up. Here are two bits I enjoyed.

• When young, Ramanujan played Goats and Tigers, a traditional game in India in which:

  Three "tigers" sought to kill fifteen "goats" by jumping them, as in checkers, while the goats tried to encircle the tigers, immobilizing them. (page 18)

  
  
• It is so hot traveling through the Red Sea that cabins on the east side of the boat, which can cool down away from the heat of the afternoon sun, are quite a bit nicer than those on the west side. Hence the acronym POSH for Port Outward, Starboard Homeward (page 199).
  

I've bought both these books to give to my young friend Artemis. (If you prefer giving games as gifts, I highly recommend Blink (under $10), Set (under $15), and Blokus.) I'm always searching for ways to enjoy the holiday spirit and still consume less. Used books are part of my solution to that conundrum. And I like getting them from Better World Books because of their interest in global literacy and taking care that old books don't end up in landfills.

May your holidays be peaceful.

Blog Heaven

• Would you like your best blog post ever (BBPE) to be in a book?
• Will you still be blogging in a year or two, when people are reading your BBPE in that book?
  

If you write about teaching or learning math, or about how people learn, and you said yes to those questions, then send me a link to your BBPE.

I'm working on a book with the (way too long) current working title of Learning Math Outside the Classroom and In: Stories from Math Circles, Homeschoolers, and the Internet. I don't have a publisher yet, but about 15 authors are working with me, each writing a chapter about the cool stuff they're doing. It's moving a bit more slowly than I'd expected (not surprising), but it keeps looking better all the time!

The book will have 5 sections:
• Math Circles, Clubs, Centers, Festivals, and Salons
• Homeschoolers and Unschoolers Do Math
• The Internet is Changing Our Lives: Teaching, Learning, and Doing Math with New Resources
• Classrooms
• Issues: Race, Gender, Gifted Kids, Public Policy (etc)

The internet section will have a few in-depth chapters, but I recently realized that 'chapters' won't do it justice. We need a compilation of BBPE. I'll be going through old editions of Math Teachers at Play, but you can save me the effort of digging out your gems by sending them in. If you're on my blogroll, you probably have a post that would work. Please don't be shy.

If you'd like to point me to anything else that ought to be in this book, I'd be delighted. If you'd like to write a chapter, that may still be possible. (I do not have enough material for the classrooms section yet.)

Email me at mathanthologyeditor on gmail etc. with links, comments, questions, and ideas.


[The fine print: A few authors have asked who gets the money. All authors of full chapters, including me, will equitably split any profits, but profits are minimal for books like this. No one is expecting to make much money, if any. It's glory for math that we're after.]

Tutoring News

My weekly tutoring gig with "Artemis" is going well. When we started, he didn't know much algebra, didn't really understand distributing. He's learned distributing mostly through experimenting, and has learned an amazing amount of algebra through playing with his TI-84. (I wrote about the first time we played with the calculator here.)

He loves coming in with graphs on his TI and getting me to guess the function. Here's the one he started with last week. (I got it. Can you?) I showed him some similar graphs to see how well he could do. He got them all. (I told his mom I figured we were into what would normally be the third year of algebra, after about two months of untutoring.)

I never have a lesson prepared for him, and we often go into college algebra sorts of topics (it's called math analysis in high school), riffing off those graphs. Some weeks I worry that we won't have a path in to good topics just by our inspirations. But it always seems to work out that we do something solid.

I know all kids wouldn't move this fast, but to me this demonstrates the power of play. That's all we're doing, playing with ideas...


My Other Student
R comes for tutoring about once a month. (I think it's a special treat for him.) He and I have been playing with Kenken and logic puzzles. He decided last week that he wanted to make his own Kenken puzzle. Yeay! That sounded like something Maria Droujkova would have suggested, but that I don't think of so readily myself. So we did it. We started with one box, and figured out what had to be true. Our puzzle isn't standard because we didn't give a total for many of the boxes. If you think about what has to be, you'll figure them out.

Here's the puzzle R and I made...

[I used Excel to make it pretty like this, and used print to save as a pdf. I learned that my blog doesn't want pdf's - it wants jpegs. So I saved the pdf as a jpeg. I don't see a jpeg option in Excel, so that might be the best way to do it...]

Anyone else want to try their hand at making their own kenkens?

Math Stories

I posted, back in June, about my favorite math books. They're ordered from youngest to oldest readers, and the first one is Quack and Count, by Keith Baker:

This is a board book, so it's good for the youngest child who will sit and listen to a story. But it stays good because it's so luscious. Great illustrations, fun rhythm and rhyme, cute story, and good mathematics. 7 ducklings are enjoying themselves in every combination. “Slipping, sliding, having fun, 7 ducklings, 6 plus 1.” (And then 5 plus 2, etc.) It would be great to have a book like this for all the number pairs that make 8, and one for 9, etc.

I teach the 'big kids' (8 to 14, most 10 and under) at Wildcat, the 'freeschool' my son attends, and another parent teaches the 'little kids' (5 to 7). She has to move from one house to another this week, and I just agreed to sub for her. I have enough trouble getting myself to move down from my college professor level to teach the 'big kids' well. I had a moment's panic when my son said, "Do easy things, Mom." Then I remembered Quack and Count.

I have 7 rubber ducks I borrowed from my son, so we can act out the story with the ducks. And then... Hmm... We have lots of active boys, and maybe a number story like this about cars would appeal to them. Maybe if I wrote it, they could illustrate it. I know they aren't big on extended writing, but maybe they'd get into thinking up other number stories.

Here's my first draft:

Crash and Count!

5 little race cars in a row
Count those race cars as they go

Racing fast and having fun
5 little race cars, 4 plus 1

5 little race cars, 3 plus 2
Looked like a crash, but then some flew!

Now they need to miss the trees
5 little race cars, 2 plus 3

5 little race cars, 1 plus 4
Most of them have crushed their doors

Turn off the engine, climb out fast
5 little race cars stop at last


[Edited to add: I just realized that this story verges on plagiarism. I copied the pattern given by Quack and Count as closely as I could. That seems totally cool for creating math lessons, but maybe not so cool for a story I'm posting online. So consider this a take-off on Baker's work.]

Number Logic

Jonathan, over at JD2718, has posted a logic puzzle he created, using lies and truths about numbers. I enjoyed it. But then our discussion turned to trying to create them.

I worked for a long time, starting from the answer I wanted and a bunch of things that were true and false about it. I couldn't get it narrowed down enough using just 5 clues. So I tried another number, and another. I think I've finally got a puzzle that works, but it doesn't feel as elegant as Jonathan's. Tell me if you get one possible answer. (But don't post the answer, please.)

And I'd love to know if anyone else has any success making these up.

Who am I?

There are four true and four false statements about the secret number. Each pair of statements contains one true and one false statement. Find the trues, find the falses, and find the number.

1a. I am the sum of two squares.
1b. I do not have any repeating digits.

2a. I'm even.
2b. I have exactly two digits.

3a. I'm prime.
3b. I am one less than a triangular number.

4a. I have just one digit.
4b. I am the product of consecutive primes.

Richmond Math Salon

Saturday, December 12
2 to 5pm

• All ages welcome. (Fun for kids 3 to 90.)

• Explore math in a fun, safe environment, where no one will judge you.

• A family math event: You and your kids can explore math in a way that works for each of you.

This month we’ll be making snowflakes, talking about symmetry, and playing with logic puzzles.

This monthly event is currently held in a small home (in Richmond, CA), so please RSVP if you plan to come. The Richmond Math Salon is hosted by Sue VanHattum, a math professor at Contra Costa College.

Interested? Email me at suevanhattum at hotmail, or call me at 510-236-8044 (before 8pm).

Schedule of salons for 2010: Jan 23, Feb 20, Mar 20, Apr 17, May 15, Jun 12, Sep 18, Oct 16, Nov 13, Dec 11.

Why I haven't been posting...

I use an iBook G4 laptop with DSL connection at home. A week and a half ago, I finally said yes to one of those pesky 'please let us update your software' messages I get regularly from Apple. My computer got trashed.

I will probably buy a new computer, but haven't done so yet. I think I'll be buying a desktop (mac, still), and getting my laptop cleaned up, fixed up, etc. Meanwhile I'm limping along, logging in wherever I can get wireless connection, since my ethernet connection doesn't work.

Anyone who wants to discuss this with me is welcome to phone me at my cell phone: 510 367 8 zero eight (one more than four). More gory details below. I've been spending more time reading actual books, hanging out in my yard, and cleaning house since the mini-disaster. It's all good, but I need to get this settled...

Today I'm seeing tons of good math posts, and am wishing I could join in more fully. But there's a 2 hour limit on my connection here, so I'm trying to just catch up on my blog reading. When I do come back, I'll have a few posts I've been working on offline. ;^)



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Gory details:
Apple techies graciously spent hours with me, even though I have no tech support contract, after I almost started crying about it being Apple's fault my computer was trashed. They and I agree that software cannot destroy hardware. But they think my hardware is at fault. (My ethernet connection broke back on April 14, and I bought a usb ethernet adapter and got my taxes out on time. So I use a usb port for my ethernet connection. One techie talked about the problem with the ethernet connection 'migrating' to the usb ports. Huh?!) My hard drive is almost full, which may have contributed to the problem. Besides no capacity for etherternet connection, I can't print. Etc

When Too Much Becomes Too Little

Joseph Ganem wrote a great article for the APS (American Physical Society) News, titled "A Math Paradox: The Widening Gap Between High School and College Math". Ganem is a physics prof at Loyola University Maryland, and works with incoming students at orientation. He's also the father of 3 children (aged 14, 17, and 20), and helps with their math homework.

I want to share 3 quotes from his article, and then encourage you to follow the link to read the whole thing.

He satrts here:

We are in the midst of paradox in math education. As more states strive to improve math curricula and raise standardized test scores, more students show up to college unprepared for college-level math.

After some examples from his kids' homework, he has these questions:

So if eighth graders are taught math at the level of a college sophomore why are graduating seniors struggling? How can students who have studied college level math for years need remedial math when they finally arrive at college? From my knowledge of both curricula I see three problems.

I'll let him tell you the 3 problems, so I don't steal all his thunder. His conclusion seems just right to me: (President Obama, Secretary of Education Duncan, Are you listening?)

All three of these problems are the result of the adult obsession with testing and the need to show year-to-year improvement in test scores. Age-appropriate development and understanding of mathematical concepts does not advance at a rate fast enough to please test-obsessed lawmakers. But adults using test scores to reward or punish other adults are doing a disservice to the children they claim to be helping.

It does not matter the exact age that you learned to walk. What matters is that you learned to walk at a developmentally appropriate time. To do my job as a physicist I need to know matrix inversion. It didn’t hurt my career that I learned that technique in college rather than in eighth grade. What mattered was that I understood enough about math when I got to college that I could take calculus. Memorizing a long list of advanced techniques to appease test scorers does not constitute an understanding.

Please read his article in all its glory, over at APS.

Help! Questions about You Tube...

Back in May, my algebra class humored me with a discussion about what made our class work as well as it did. One student did a video of the discussion. She uploaded that onto YouTube and then deleted it from her camera. YouTube rejected it because it was too long. Is there any way I can recover that? I had assumed not, but I see one scene from it on the list of videos, along with the notice that it was rejected.

Also, can I moderate or prevent comments, or delete them? My students were brave enough to get videod while explaining math problems, and now someone has made a nasty comment.

Math Teachers at Play #19

Are you wondering where MTAP #18 went? Here's the story (contest-winning entry from Lisa Downing), and we're sticking to it!

The Odds were at odds with the Evens. It never seemed fair to them that two Odds made an Even but two Evens didn't make an Odd. Fifteen fired the first salvo by stepping into the order twice. Sixteen managed to jump in, but then Eighteen disappeared. Seventeen and Nineteen were prime suspects. The Numerologist stepped in and told everyone to get back in the right order or ELSE. Unfortunately Eighteen was still missing. The authorities launched an investigation but there were so many factors involved that they never could get to the root of the problem.

Riddle: What do 10011, 23, and 13 have in common?

Gorgeous! The photo above, of an anglerfish ovary magnified 4 times, was taken by James E. Hayden, and won 4th place in the Nikon Small World photomicrography competition. You can see lots more winning photos here. More about Hayden at the end.


Getting down to business, allow me to welcome you to a math buffet. I think there will be something here to tickle everyone's palate. (Photo by skenmy.)

• Maria Andersen is at it again! She's redesigned her math for elementary teachers course, and shows us some of the results in Transforming Math for Elementary Ed. This post is full of links to work her students did, some of it exciting stuff.
• John Golden offers us a game called Area Block modeled on Blokus (one of my current favorites to play with kids). I haven't had a chance to play it yet, but I am definitely looking forward to it. (Here's a bit of math teachers at play trivia. John and Maria work within 30 miles of each other in my home state. What state is it?)
• Denise walks us through an algebra word problem, translating from English to "mathish". Nice! And Jason wants our help with teaching students how to read a few different types of problems, in "When vocabulary isn't the issue" and "A reading experiment". The puzzle given in that second post looks fun!
• Pat Ballew gets us thinking about geometry with his "Notes on Cyclic Quadrilaterals".
• Do we discover math or invent it? How we answer that question affects how we talk about the math of the ancients. Dan MacKinnon reviews a book and discusses this intriguing issue.
  
• A short review of The Little Book of Mathematical Principles is at We Overstep.
• Pumpkin Patch offers us another game, Sum Math.
• The descriptions don't always look accurate to me, but you may find some goodies among the "100 Incredible Open Lectures for Math Geeks". Some are audio only, some are videos.
  

If you haven't stuffed yourself yet, here are a few more tidbits I noticed over the past two weeks.

• Two mysteries solved, one a murder, and one the opening chord of Hard Day's Night.
• Sometimes there's a hard way to solve a problem, and an easy way...
• Does a bullet that's fired from a gun land at the same time it would if it were dropped? Built on Facts gives us some equations and Myths Busters weighs in on YouTube.
• On thinking before you compute.
• Like comics? Indexed on confusion as a function of information, and a graphic novel about Bertrand Russell.

_____________

Here's the interview Nikon did with James E. Hayden. I wrote and asked him how he uses math in his work. He said:

As for the math - it is, of course a large part of doing any kind of research. In a lot of my regular work, we work with analysis programs that quantify different aspects of the images we capture. Everything from simple counts (how many cells in the field of view?) to time lapse analysis - how fast are the cells moving? In what direction? At what angle to the original movement? Does the rate of movement change over time? Do we need to quantify the interaction of the cells in some way? Is there a change in the fluorescence intensity values? And other interesting things like that. My assistant, Fred Keeney (who won an Image of Distinction in this year's competition) has been taking computer programing classes to help us automate these kind of analyses.

Submit your blog article to the next edition of math teachers at play using our carnival submission form. Past posts and future hosts can be found on our blog carnival index page. (Our schedule is changing to once a month. Denise at Let's Play Math! will host the next carnival on November 20.)

Contest: Write a number story by Wednesday

The Math Teachers at Play blog carnival came out twice as #15. Since then we've had #16 and #17. We'd like to iron out the numbering, and so the upcoming issue will be #19.

I am personally sponsoring a contest for the best little (ie, very short) story written about how the numbers got mixed up this way. The winner gets their story included in the next MTaP, which comes out here on Friday, and gets a $10 gift certificate at Better World Books. I get to be the judge. :^) It could be funny, mysterious, intriguing, whatever will be memorable.

Deadline: Midnight (PST) on Wednesday

Why? Partly to have fun with our glitch, partly to 'make math our own' as Maria Droujkova likes to say. And partly because I came to math on the internet through 'living math' - Julie Brennan's (trademarked) term for math through stories, and for the list she runs for mostly homeschoolers and the site she maintains with gazobs of ideas for how to engage kids in mathematical thinking through math stories and finding math in any story.

The Internet is a big treasure hunt!

I'm laughing at myself right now. I wonder if I can make this funny for you. Before I explain I should tell you ... my memory is so bad... (I once got a card that had an old woman saying that on the front. Open it and see, "How bad is it?" in a bubble. She replies... How bad is what?) And maybe I should claim that my brain takes a bit to get into gear in the mornings?

Last Thursday I posted a bunch of links, and included:

I can't remember now why I did this, but I'm still intrigued... I went to Wolfram Alpha and typed: factor 1782^12+1841^12. It's just a bunch of big numbers. Why do I like it?

On Saturday night Joshua Zucker replied:

why 1782^12+1841^12? I don't know why to factor it, but of course it equals 1922^12 (just try it on your calculator, not at Walpha of course!)

Well, I wasn't thinking of the consequences, and I believed him. I was impressed that he could use his calculator to factor the huge number you'd get from 1782¹²+1841¹². I didn't reach for my calculator, being comfortably ensconced in my recliner. No, I reached for Google, and I googled Josh, because I was curious about what math he might lead me to.

I found a comment he made on Cut the Knot many years ago (in 2000). So I started exploring Cut the Knot, and thoroughly enjoyed some discussions about how math uses words differently from their common usage.

Eventually I remembered my original quest and googled "1782^12+1841^12". The very first thing I got was:

Fermat's last theorem. Statement that there are no natural numbers x, y, and z such that x^n + y^n = z^n, in which n is a natural number greater than 2. ...

[Fermat's last theorem was proved in 1995 by Andrew Wiles. There are lots of whole number solutions to x² + y² = z², like 3² + 4² = 5². But the theorem says there are no solutions to x³ + y³ = z³, nor to equations like that with higher powers.]

Huh? But Josh said 1782¹²+1841¹²=1922¹²?? Next entry I clicked on was about a Simpson's episode:

In the 1995 Halloween episode of the award-winning animated sitcom The Simpsons, two-dimensional Homer Simpson accidentally jumps into the third dimension. During his journey in this strange world, geometric solids and mathematical formulas float through the air, including an innocent-looking equation: 1782¹² + 1841¹² = 1922¹². Most viewers surely ignored this bit of mathematical gobbledygook.

On the fan discussion site alt.tv.simpsons, however, the equation caused a bit of a stir. “What’s going on, he seems to have disproved Fermat’s last theorem!” one fan marveled, referring to the famous claim by Pierre de Fermat—proved just months earlier—that for any exponent n bigger than 2, there are no nonzero whole numbers a, b, and c for which a^n + b^n = c^n. The Simpsons equation, if correct, would be a counterexample to the theorem, meaning that the proof had been wrong.

Ahh, now I get it! And I finally had a vague memory of reading a post somewhere about how 1782¹² + 1841¹² and 1922¹² look exactly the same if you evaluate them on a calculator. (Try it!) That must be why I had originally gone to Wolfram Alpha with this.

Meanwhile, here's the other treasures I found:

• A silly math discussion on nrich
• On simpsonsmath.com, Lisa Simpson inspires Girls Just Want to Have Sums
• I learned how to use html to do the powers right (angle brackets around sup and /sup, before and after)
  

Is anyone else giggling, or is the humor lost in translation?

Math Education Research

Education is a complex and messy endeavor. We all have our own ideas about how children should be raised, about how learning happens, and about what is important for children to learn. The schools have to deal with parents on all sides of every political spectrum demanding what they think is needed for their children.

Educational research that doesn't acknowledge this messiness, that tries to buy 'scientific' cachet with control and treatment groups but frames the questions too narrowly, is more likely to reinforce the values of one group than to deepen our understanding of the learning and teaching enterprise. (Of course, we're each more likely to see flaws like this in research that doesn't resonate with our own values.) In this post, I want to dissect a flawed study, together with my friend Ben, and then give links to some studies I've enjoyed reading. I would actually appreciate it if any of you would like to point out flaws in some of those. (I'll start a new post for each one, so we don't get all tangled up.)

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“The Advantage of Abstract Examples in Learning Math”, a study done by Jennifer A. Kaminski, Vladimir M. Sloutsky, and Andrew F. Heckler, was published in Science magazine in April 2008 (you need a paid subscription to see the article) and highlighted in the New York Times soon after. I had responded to this study before I started blogging, in email to colleagues. It was brought to my attention yesterday by Ben Blum-Smith's new blog, Research in Practice, where he writes a great critique of it. I agree with all he says (here and here) and want to add a bit more.

The people who did the research, along with the unquestioning NYT author, say that the research shows that students learn math better with abstract examples than with concrete examples. Ben and I are saying that their research design is severely flawed, and that they've shown nothing useful.

Ben gives a great description of what the research folks were supposedly trying to teach, which was the properties of a "commutative mathematical group of order three". That's a fancy name for something not much more complicated than clock arithmetic. Imagine a clock that only has three hours on it, and instead of a 3 at the top, it has a 0. So 1+1 is still 2, but 1+2=0 and 2+2=1. This is the most common example used when people are first learning about these groups. The examples used in the research seem very contrived in comparison.

Both the research folks and Ben described a tennis ball factory, where you're keeping track of how many balls you have in hand, after you've put as many as possible into those 3-ball cans, but Ben's description makes a lot more sense than the one used in the research. (When I originally read this study over a year ago, I never saw the tennis ball example. In the one 'concrete' example I was able to find details for, they used a full cup for the 0, or identity element, which I would find confusing.)

People who actually study groups like this sometimes do it without numbers. The 'elements' of the group might be labeled a, b, c instead of 0, 1, 2. There are properties that can be studied, like identity elements and inverses. (0 is the identity because adding it to other elements doesn't change them. 1 and 2 are inverses because 1+2=0, the identity. These properties can make sense even when the elements aren't numbers.) So the researchers 'taught' this using 'abstract' examples for some subjects and 'concrete' examples for others. They quizzed all of the subjects using a group consisting of a vase, a ladybug, and a ring. Although concrete, these strange elements fit much better with the 'abstract' example than with the 'concrete' example. It's not surprising that the subjects whose example was more similar did better when quizzed.

There is lots of narrowly focused research like this out there. It may be useful in physics to narrow a question down to one detail, when the interactions between the small parts is clear. But in social arenas, all the parts interact, in very complex ways. Research like this cannot tell us much of value, even when its design is less flawed.

Kaminski et al want to say that their research tells us children learn math better without concrete examples. Their claim is very political. To promote it, they have done a number of studies with minor variations. (Googling their names, I see work done in 2003, 2006, and 2008.) I'd rather see education research that addresses the big, messy picture. Here's the MAA president-elect's take on this, and here is an article interviewing one of the 3 authors of the study.


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Research that I've found more interesting is much broader. It doesn't attempt a double-blind statistical power, which can only come through narrowing the questions until they become too artificial to be of use.

I've been reading Jo Boaler's work on the benefits to students at all skill levels of working together with one another. On first glance, it may seem that tracking would allow the best students to go farther, and allow slower students to get a more solid grasp on what they're studying, but tracking actually harmful to students at both ends and those in between. Boaler shows why, and shows how to make heterogeneous grouping work. One article is here, or you can read her book, What's Math Got to Do with It?

Alan Schoenfeld wrote the book Mathematical Problem Solving, in which he describes his very detailed research into the process followed by math students versus mathematicians while attempting solution of a hard problem. He taught a course in problem-solving strategies, using Polya's framework, and gave a pre-test and post-test to those students. Compared to students taking a more typical math course, these students' problem-solving skills improved significantly more. Here is an article of his on a different topic, how mathematical conversation in the classroom promotes learning.

Both Boaler and Schoenfeld compare groups with and without the 'treatment' they believe is effective, and show evidence for their belief, as Kaminski et al did. There are other sorts of research, which attempt to understand children's learning processes, but which don't have 'control groups'. One such project I'm interested in following is Measure Up, which introduces math through measurement and algebraic reasoning. Here's an article from that project.

Our biggest problem may not be understanding better how children learn, but implementing the good ideas that come from this research. When most elementary teachers are uncomfortable with mathematics, what we need to focus on is how to help them. Liping Ma's book, Knowing and Teaching Elementary Mathematics, compares elementary teachers in the U.S. and China. The Chinese teachers understand the math much more deeply. A good summary of the book is here. And an article by Ma is here. And here's a piece by another researcher, Hung Hsi Wu, on the depth of understanding needed by elementary teachers.

What math education research have you found valuable?

What math is used for...

When I first got out of college, in 1979, I wanted to do something with my math degree besides teaching. I knew I wanted to eventually become a teacher, but I figured I'd be a better teacher if I had a deeper understanding of what math is used for.

I think I had an interview at that time at Bechtel. (It was someplace with more security than I'm accustomed to.) I know I thought a lot about how most of the employers who might want my talents were doing things I didn't approve of. Code-breaking sounded fascinating, but back then code-breaking meant working for the government (or so I thought), and I was no happier then than I am now about my government's warring tendencies.

I ended up doing some computer programming for a very small company. It was business reports - not very exciting, but nothing terrible, either. After a bit more than a year, the company folded, and I headed toward teaching.

I haven't had to think about this issue much in the 30 years since then. I've seen much broader uses of math, and coding-breaking has come to be identified more with information security than with espionage. But of course the war-makers are still big employers of mathematicians, and that was made clear to me this morning by a post that looked really fun at first.

Liz, at STEM-ology, posted It's a Math World, After All, about a cool new 'ride' at Disney World called Sum of All Thrills, that lets kids (and adults?) design their own ride, and then "experience it on a giant robotic arm simulator." (It reminded me of the turtle geometry I was just reading about in Mindstorms.) I loved it!

But then I followed her link to the original Yahoo article, and saw this:

"Sum of All Thrills" sponsor Raytheon has nothing to offer the average consumer. But the high-tech defense and homeland security contractor does have jobs for those passionate about engineering...

I wasn't planning on going to Disney World any time soon, but Disneyland was the high point of a big trip my family took when I was 12, and I might be willing to take my son some day. This is a reminder to me of how much corporate propaganda is built into places like that. My comment on Liz's blog ended with "That's a show-stopper for me. War-mongers get way too much access to our children, and I have a problem with that..."

Have any of you struggled with math's less wholesome uses?

Math Teachers at Play

Dan has posted Math Teachers at Play 17 at his mathrecreation blog. (There were two issues of MTAP #15, so this is really the 18th issue.) I've been negligent about linking to each issue of MTAP, so here's an archive of links to all the past issues (click on 'past posts').

Two of the posts I especially enjoyed:

John Cook has posted on the math behind musical scales in his posts Circle of fifths and number theory and Circle of fifths and roots of two at his blog The Endeavor.

Alison Blank has put together an inspired and inspiring Prezi presentation, Math is Not Linear, and posted about it on her blog Axioms to Teach By.

I added quite a few blogs from this issue to my already long list at Google Reader. I think I'm really going to like the quirkiness of this blog:

Vlad Alexeev shows us an impossibly small book of impossible figures in the post Mini Books of Anatoly Konenko at his blog Mathematical Paintings and Sculptures.

Next issue is in two weeks, right here. After that, it looks like we're switching to a once a month schedule, on the 3rd Friday of each month (with Carnival of Mathematics taking 1st Fridays).

Blog Action Day - Climate Change

Today is Blog Action Day 09 - Climate Change. (Thanks for the pointer from squareCircleZ, who wrote a great post on it.)

What's that got to do with math? Well, we won't have time for the cerebral pleasures of math if we're dealing with floods and droughts, famine and war.

Here's a site that give some numbers to help you think about it. I try to live simply, but if everyone lived like I do, we'd need over 2 planet Earths.

I wanted to write more, but this is already coming out late in the day...

Links on Thursday

I've been saving these up, I guess ...

I keep hearing about thought experiments in physics lately. Here's a good post on that idea.

This post is mostly about John Conway, the game of life, and how that relates to segregation (watch the video).

Here's a post on the n-queens problem. (For n=8, put 8 queens on an 8x8 chessboard so none are attacking any others.) On Sunday evening I played around, trying to find a solution, and couldn't. On Monday morning I showed the problem to Artemis (the boy I tutor), who said, "That has 92 solutions, unless you don't count rotations and reflections. Then it has 12 solutions." I wanted to get past his memory and work with thinking about it. He put 4 queens on a graphpaper board, and said "now go from there". I told him I'd done that much and gotten stuck. He and I crossed out places queens couldn't go, and I suddenly saw a solution. I told him he was a good teacher. [He also showed my his attempt to multiply out (A+B)^5. He isn't yet grounded in why you do the steps you do; he did all the right steps plus a bunch more. We'll get there... :^) It sure doesn't feel like work, tutoring him.]

This one is not really math. NASA is planning to crash something into the moon tomorrow, and it should be visible with garden-variety telescopes. Lesson plans available, and I might do something with the kids I teach, if I can pull it together in time.

Tanya Khovanova doesn't like IQ tests, and has made a funny question up that she's pretty sure won't appear on anyone's IQ test.

This woman knows how to take learning into her own hands! She had a concussion, and decided to create a multi-player role-playing game called SuperBetter to help herself recover. Wow! (Thanks, Dan, for pointing me there.)

Sweeney Math has a nice systems of equations project for an algebra II class, or a stat class. "Students find data online that they are interested in comparing. (Sales of video games v sales of movies, Wins of their favorite sports team v wins of their friend's favorite sports team, Women's race times v Men's race times, Success of movie with many sequels v another, Sales of Abercrombie v sales of American Eagle, etc) They graph and find best fit lines for each set of data, then answer some thought provoking questions about the results." One of the questions is the point of intersection and what it means. I like it.

At God Plays Dice, there's an interesting (to me) book review post titled Counterexamples in X, where X is a field in mathematics.

I don't have an interactive whiteboard available where I teach, but I do have a 'smart classroom' (computer hookup and internet projection). Most of the examples in this Interactive White Boards interview can be used in my classroom, I think.

I can't remember now why I did this, but I'm still intrigued... I went to Wolfram Alpha and typed: factor 1782^12+1841^12. It's just a bunch of big numbers. Why do I like it?

Oops! There was one more. I remember from when I was young, a letter to Ann Landers, claiming that you get wetter in the rain when you run than when you walk. Here's a good physics analysis debunking that.

Mindstorms: Children, Computers, and Powerful Ideas

I have a new hero. Seymour Papert writes so brilliantly about math, learning, and how it all fits together, I think I'll have to read his book a few times to absorb it all. He wrote Mindstorms: Children, Computers, and Powerful Ideas back in 1980. (Why I never read it until now is a mystery to me. I've taught programming and math since the early 80's and could have used these ideas.) I expected a book about computers from the 1980's to be pretty severely dated, but the ways in which it's dated are surprisingly trivial. Papert's notions about why programming a turtle is valuable are still true, powerful, and not widely applied. But the book goes way beyond programming turtles.

He starts the book with a story from his childhood, about how he was in love with cars, and at two knew about "the parts of the transmission system, the gearbox, and ... the differential" (more than I know even now). He adds:

I became adept at turning wheels in my head and at making chains of cause and effect: "This one turns this way so that must turn that way so..."
Gears, serving as models, carried many otherwise abstract ideas into my head... I saw multiplication tables as gears, and my first brush with equations in two variables (e.g., 3x+4y = 10) immediately evoked the differential. By the time I had made a mental gear model of the relation between x and y, figuring how many teeth each gear needed, the equation had become a comfortable friend. (page vi)

But of course not all children will fall in love with gears the way he did, hence his "attempts ... to turn computers into instruments flexible enough so that many children can create for themselves something like what the gears were" for him. He points out many ways in which the gears encouraged his understanding of mathematics, including affect (he loved them), body knowledge (he could turn his hand or body the way the gear turned while he was thinking about it), and flexibility as a model for mathematical structures. His creation on the computer, the LOGO language, included a turtle on the screen (or a robotic turtle) that could be moved around.

He worked with Piaget for years, and has a similar clarity about the deep learning that must happen for children to understand things that seem very basic to us adults. He has differences with Piaget, though, and the most salient here is his conviction that the cultural environment makes a difference in when kids will learn things. To learn formal systems like mathematics, it helps for kids to have a fun "world" to play in that uses formal systems, like LOGO. So Piaget saw the 'formal reasoning' stage of development happening around 12, and Papert thinks much younger children can do formal reasoning if given the right environment.

He has a lot to say about the damage wrought by the culture associated with schooling:

Our children grow up in a culture permeated with the idea that there are "smart people" and "dumb people". The social construction of the individual is as a bundle of aptitudes. There are people who are "good at math" and people who "can't do math". Everything is set up for children to attribute their first unsuccessful or unpleasant learning experiences to their own disabilities. ... Within this framework children will define themselves in terms of their limitations, and this definition will be consolidated throughout their lives. Only rarely does some exceptional event lead people to reorganize their intellectual self-image in such a way as to open up new perspectives on what is learnable. (page 43)

Of course kids in school hate making mistakes, and want to throw the mistakes away, or run away themselves. But if they're doing programming on a project they care about, the mistakes become bugs that need fixing, not testaments to their inadequacy, and they become willing to debug. The more they get into that habit, the more willing they'll be to deal with future 'mistakes' that way.

Most of us learned Euclidean geometry in high school, with its axioms, straightedge and compass, and our first taste of proofs. (There are alternates to this, non-Euclidean geometry and Origami geometry, that still use a system of axioms and step-by-step deductive proofs.) Analytic geometry uses the x and y coordinate system to connect algebra and geometry. Papert mentions those two and then talks about how turtle geometry is both easier for kids to connect with (tell the turtle how to move in a circle, by figuring out how you'd do it) and more sophisticated (it has a deep connection with calculus). Once a child has really played with turtle geometry, they're likely to feel more at home as they learn about other geometries. Papert goes into how using turtles to think about physics is likely to lead into some deep science learning, too.

Reading Mindstorms motivated me to find and download Scratch, a modern descendant of LOGO, and start learning it. Scratch has 'sprites' instead of the turtle. You can create as many sprites as you want, and give each one a script. This week I've brought my computer in to Wildcat, where I teach kids in a very free-form environment, so they can play with Scratch. They are loving it. I'll probably post soon about that.

While I was online, searching for more information about Papert's recent work, I discovered that he'd been in a tragic accident. While in Hanoi in December 2006 for a conference, he was hit by a motorbike and suffered a severe brain injury. There is hope he will eventually recover, but he hadn't yet as of July of 2008. Here's the news article from then. I've searched and haven't found anything more recent. I'm wishing him well.

I want to include so many quotes, but I think I'll just write more posts on this later. If you want to think deeply about how children (and adults) learn, read this book. If you want a fresh perspective on how computers might be used with children, read this book. If you want more reasons to shake your head over the current testing craze in the public schools, read this book.

Why Math? Why School?

Part I. Why Math?

Deborah Meier (author of The Power of Their Ideas) co-writes the Bridging Differences blog with Diane Ravitch. Meier's topic today is 'Why School?' She is discussing what she hopes students will learn in school. Deborah Meier has done amazing work, and I usually like what I read at her blog. But her conception of math seems terribly shallow to me:

Sufficient mathematics to make sense of what they find in the media—statistics, probabilities, forms of graphing, percentages, et al to a high degree of sophistication by the time they are 16. Basic arithmetic computation by 13.

I agree that one reason to learn some basic math is to be able to have intelligent opinions about national issues: Is it more costly to have single payer health care or what we have now? Is social security doomed because more and more of our population is the elderly? I would never have expected to be interested in a blog on the tax code, but Mary O'Keefe (who runs the Albany Area Math Circle) writes great posts about issues like this on her tax blog. How can you understand national budget questions if you get nervous about numbers?

So yeah, these are reasonable goals, but dreadfully insufficient.

Here's what I wrote in response:

Deborah, I have a deep respect for you and the work you've done. So I was distressed to see your opinion of what math students should know - mostly arithmetic and statistics. Well, that's a fine start, but it is not enough.

Shouldn't they know enough math to understand science? Shouldn't they see the beauty of math? Two books, accessible to anyone, that I'd highly recommend, are The Cat in Numberland, about infinity, and Powers of Ten. (I've blogged about a number of fun math books at my blog, Math Mama Writes.)

What Diana said above about literature and history applies to mathematics as well: We will teach mathematics because it is important and beautiful. We will teach it not because it will save our society, not because we "must" know particular techniques, but because we simply do not have it in our hearts to do otherwise.

I don't feel like I was particularly eloquent. If you think math is important for more than these basic uses, please go on over there and say your piece.

And I'd love to hear from you here. Why is math important? And what math is vital for schoolchildren to learn? I love math, but I don't feel clear on why people who don't love it should learn it (beyond the basics discussed above).


Part II. Why School?

Meier labeled her post 'Why School?' in response to Mike Rose's new book with the same title. I've enjoyed his previous books, Lives on the Boundary and Possible Lives, so I expect to like this one too. Whether it will answer some of the questions I find most difficult remains to be seen.

My personal vision of the ideal school is more like a kids' community center, where the children decide how to spend their time, and are surrounded by resources and adults who want to share in the learning adventure. Deborah and her respondents talk about what should be 'required'. I don't think it's possible to require students to learn anything. The best we can do in a system based on requirements is to show the students (who are usually still eager to learn, if it appeals to their own values and priorities) why our subject is vital. That's why I love the work Dan Meyer and Kate Nowak are doing making high school math topics relevant for their students.

These two questions go deep for me: Why Math? Why school?

The Joy of Tutoring

Artemis* is 8. He arrived for his first tutoring session last week ready to learn more about trigonometry. He doesn't yet do algebra, and a few days ago said "I can't subtract", but trigonometry is what he's enamored of right now. (He can subtract, he just doesn't know how to use the standard algorithm, yet.) He's full of extremes like this. He was reading before he was 2, but had very little control of his body until recently.

For the past year he's been coming to the math salon I host, along with his parents and twin sister. The first time he came, he was so excited he just had to twirl around and get his whole body moving. (He reminds me of myself. When I'm really excited, I just have to wiggle and wag my tail.) He's still excited, but now he can be part of a group of people working together on a math problem. Part of his excitement during our tutoring session was that he got to have me all to himself. He snuggled up next to me on my sofa, and we dove in.

I started with the Pythagorean Theorem. He knew there were hundreds of proofs, but I don't think he'd really walked through one before. The proof I'm most familiar with involves a bit of algebra, and for him that was the complicated part. (Maybe he's ready for lots of heady stuff but not yet algebra? We'll see.) Just now I looked up proofs to try to find the one I used. Didn't get a good link for that one, but here are two I'll show him next Monday, both completely visual: One with the triangles hinged, the other with them sliding.

[In a previous post I mentioned mathematical holes that can cause students grief for years and years, like not learning your times tables in 3rd grade because you were out sick. I was unsure whether I wanted to say that because Artemis and others like him were in the back of my mind somewhere. When a student isn't expected to know things in a particular order, it's not too hard to work around them, and get to them later.]

I showed Artemis a few more basic geometry proofs, like the angles in a triangle adding to 180 degrees. In the middle of our one-hour lesson, he got so excited by it all he just had to move, so he took a 10-minute break on the trampoline. At the end of our lesson, I lent him Geometer's Sketchpad, Who Is Fourier?, and Mathematics: A Human Endeavor. He's been reading the Fourier book since then, and came in this week excited about one of the formulas he saw in it.

[Ooh, this is my first time doing that. I like it! I used codecogs.]

It seemed to me that he was intrigued by the fact that sine isn't additive. So I played the mystery box (or Guess My Rule) game, where I have a function in mind, and he figures it out by giving me inputs to see what outputs I give him. It gave me a fun way to talk about functions, input, output, domain, range, etc. With each of the functions I used, I then drew a graph, and we looked at whether it would be additive. I talked about it as linearity.

He wanted to think about , so I pulled out my TI calculator. I tried to keep chatting with him, but I found myself saying "Look!" a few times, and belatedly realized he was too entranced by the calculator to do anything else. So we looked at things like , which I knew went with what he'd been reading in the Fourier book. I let him borrow the calculator, and later that day his mom went out and bought him one.

Next week he wants to take a walk and find math all around us. Sounds fun to me. (It took me a moment to let go of the notion that we had to do something more industrious.) ;^)

I am like a kid in a candy shop myself, getting to work with someone who loves math so much. It feels like jazz improv, taking his lead and doing a riff on it. Wow! I'll be taking on a few more students in the coming months. I wonder if any of the others will lead me as well as he does, so I can learn more about how to teach by following.


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*He decided to use the pseudonym Artemis for my blog posts because he likes the Artemis Fowl books.

I Love Nerds!

First off, you gotta know that I've been trying to reclaim the word nerd as something positive for years. One of my favorite bands in the 80s was The Roches. The refrain in their song Nurds has the line "Nurds, I'm so glad I am one!" And googling got me this charming link for a radio show about nerds.

Maria Andersen just posted this Calculus Rhapsody video on her blog, and I'm still giggling! (I've linked to YouTube. If that's blocked where you are, check out Maria's embed.)

So are there any fun, complimentary words for us nerds? (None found in the online thesaurus I just checked. In fact, nerd is connected to stupid person and fool there more often than anything else.) Our culture clearly has issues with the power of intelligence...

Math Relax: A Guided Visualization for Overcoming Test Anxiety in Math

If you are nervous during tests, try listening to this every night for a few weeks. (Although I don't feel the quality is perfect, and eventually hope to redo it, many students have found it very helpful.)

If you like it, please let me know.

Click Here for Soundtrack

This recording combines relaxation techniques with a guided meditation focused on enjoying math (and tests) more. It may seem absurd now, but if you repeatedly imagine yourself actually looking forward to a test, then you’ll eventually find your outlook at test time to be at least a little bit more positive. The techniques I use in this recording are taken from the work of Margo Adair in her book on applied meditation, Working Inside Out.


Credits
Voice: Sue VanHattum
Flute music: Wayne Organ
Applied Meditation Concepts: Margo Adair
Script: Sue VanHattum
Recording Studio: Contra Costa College
Preliminary Release: Muskegon Community College
Tested by: students at Muskegon Community College and Contra Costa College


If you find this helpful, please let me know. If it has made a big difference for you, please consider making a donation in Margo Adair's name to one of the following organizations. Margo Adair died of cancer on September 2, 2010. Please make sure the organization is still active before donating.

• Community Alliance for Global Justice 
• Detroit Summer
• Seattle Young People's Project

How To Use This Recording

Before listening to this recording for the first time, read “That’s How Math Is” (below), which talks about math learning, and includes a summary of problem-solving steps, so you can approach math from a good perspective.

The more often you listen to this recording, the more effect it will have. I recommend listening at bedtime every night. (Don’t worry if you fall asleep during the recording, your subconscious will still hear it.) If possible, start at least 2 weeks before your next test. Whenever you start, keep using this recording through at least two months and two tests. Whether it’s helpful or not, I’d like to hear from you.

Contact me if you'd like a script of the recording. (Useful if you want to make a recording in a different voice, or with changes to the words, perhaps to use for a different subject.)

Jean Harvey, a student who used this while taking the Beginning Algebra course at Contra Costa College, says it took about 3 weeks of listening to it every night for it to make a difference. She didn’t expect to pass 118 and ended up earning a B in the course. She went from a 69% on the first test to a 96% on the second test.


“That’s How Math Is…”

Some things to know about learning math:
• In an ideal world, everyone would have lots of hands-on experiences to help them internalize ideas related to number, would always learn at their own pace, on their own schedule, and would have access to tutors and mentors who love math and love helping people find a good path to follow in order to learn it.

This is not that world - many students learned math from teachers who were themselves uncomfortable with it. (I’d guess about 4 out of 5 people are uncomfortable with math, and elementary teachers probably aren’t any better than the general population in this.) Those teachers were not able to explain math concepts in a way that made them make sense, and were often tense and would do ineffective things like requiring students to follow the book’s method. So the cycle of discomfort continues.

• Math concepts build on the ones before in a way that’s not seen in any other subject area. Even with good teachers, if you miss a few months in third grade (for example), that hole may cause you grief forever. If you recognize that there are holes in your past learning, it will be especially helpful to work with a good tutor or mentor to fill them in.







• Math is not about memorization; it’s about understanding. Ask why every step of the way, and you’ll learn math in a deeper, more satisfying way.

• Once you understand something well in math, it suddenly seems so easy that it’s hard to understand why it took so long to ‘get it’. This is true with any new concept, but it’s particularly noticeable with math: Imagine… You’re in class, struggling with a problem that seems impossible, and the person next to you blurts out “That’s easy!” You feel like a fool. It’s happened to just about everyone, including that person who thought it was easy. This happens partly because new synapses (connections between neurons/brain cells) are made as you learn – once they’re made the thing that seemed impossible now seems easy.

Even mathematicians are likely to feel dumb at first when looking at a new problem. That sensation of having no clue how to get started can be overwhelming. But the good mathematician has had enough successful experiences in their past that they find it easier to tell themselves they can do it. (When faced with a problem that’s hard for me, I often have this argument going on in my head: I can’t do this! Yes you can. No I can’t…)

Good mathematicians also have some techniques for problem-solving that help them break things down. Here are the 4 steps that George Polya proposed (but there's much more to it than this). More on this here.
1. Understand the problem.
2. Make a plan for how you might solve it.
3. Carry out your plan.
4. Look back. (Check your work, see how it might apply to other problems, etc.)

Solving math problems can be a real struggle, but the satisfaction once you do solve your problem can be quite powerful. Think of math problems as puzzles to solve, think of yourself as a detective, and have fun!



Added on 11/21/11: Test anxiety can be addressed in many ways. This guided visualization is one way. Googling 'test anxiety' will help you find many other ways. One method you might find helpful is described here, along with the research supporting it.